How do momentum, impulse, and GPE factor into solving physics questions?

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Momentum, impulse, and gravitational potential energy (GPE) are crucial concepts in solving physics problems, especially in competitive contexts. Understanding the relationships between these concepts can aid in analyzing motion and energy transfer in various scenarios. Participants in the discussion emphasize the importance of breaking down problems step by step to identify where confusion arises. Clarifying the application of formulas related to momentum and GPE can significantly enhance problem-solving skills. Seeking assistance and verifying calculations is essential for mastering these physics concepts.
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Hey guys I am preparing for a physics contest and I am stuck with on a few questions. they are step by step questions. If someone can clearly explain how how to solve the following questions it would greatly help. Thank you.
 

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Please show what you've done so far and point out where you are stuck.
 
this is what I've got so far but I am not sure if its right.

2jf0qdt.jpg


2gx0k6c.jpg


thats what I've figured out so far but i need to verify this.

please help.
 

Thread 'Rolling without slipping on a curved surface'

Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
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